Running head: PURPOSE OF AUDIT & AUDIT OPINION LETTER
Purpose of Audit & Audit Opinion Letter
September 30, 2016
HSM 260
Ms. Gardner
1
PURPOSE OF AUDIT & AUDIT OPINION LETTER
2
Purpose of Audit & Audi
Running head: CATALOG OF FEDERAL DOMESTIC ASSISTANCE
Catalog of Federal Domestic Assistance
September 23, 2016
HSM 260
Ms. Gardner
1
CATALOG OF FEDERAL DOMESTIC ASSISTANCE
2
Catalog of Federal Domesti
Running head: GENERALLY ACCEPTED ACCOUNTING
1
Generally Accepted Accounting Principles (GAAP)
September 18, 2016
Ms. Gardner
HSM 260
GENERALLY ACCEPTED ACCOUNTING
2
Generally Accepted Accounting Princ
Running head: APPLICATION AND ANALYSIS OF FINANCIAL TOOLS
Application and Analysis of Financial Tools
1
APPLICATION AND ANALYSIS OF FINANCIAL TOOLS
2
Application and Analysis of Financial Tools
Part 1
Running head: APPLICATION OF RATIOS TO DETERMINE FINANCIAL HEALTH 1
Application of Ratios to Determine Financial Health
October 2, 2016
HSM 260
Ms. Gardner
APPLICATION OF RATIOS TO DETERMINE FINANCIAL
Running head: A COMPARISON OF THE ACTIVITY
1
A Comparison of the Activity & Cash Flow Statements
September 23, 2016
HSM 260
Ms. Gardner
A Comparison of the Activity & Cash Flow Statements
A COMPARISON
(16.10) when (t, x) = is constant. At this point, we observe that the
formulation of the option pricing problem sounds rather similar to the
formulation of the Feynman-Kac representation theorem which
i(T t) o cos i 2 (T t) ai i sin i 2 (T t) 2ab 2 . As in
the calculation of (t), the final simplification is to note that cos(iz) =
cosh(z) and sin(iz) = isinh(z) so that expcfw_(t) = expn ab(ai)(T t)
from 0 to t. Recall that Z tan(z) dz = log(sec(z) = log(cos(z) and so (t)
= (0) abDt it abE Z t 0 tan arctan D E F(T s) ds = (0)
abDt it abE F log cos(arctan D E F T) cos(arctan D E F(T
t)! . The te
model is that geometric Brownian motion tends to increase
exponentially which is an undesirable property for volatility. Market
data also indicates that volatility exhibits mean-reverting behaviour.
T
standard Brownian motion with B0 = 0, if we can solve the SDE, then we
can determine XT () for any T 0. Example 21.1. Consider the case
when both coefficients in (21.1) are constant so that dXt = dBt
equivalent to the following statements: (i) |x y| |x| + |y|, (ii) |x + y|
|x| |y|, (iii) |x y| |x| |y|, and (iv) |x y| |y| |x|. Example
25.8. Let X = R 2 . If x R 2 , we can write x = (x1, x2). If we
k2 t = 0. (In fact, these approximations can be justified using our
result on the quadratic variation of Brownian motion.) Hence, we
conclude that d dt f(Bt) = f 0 (Bt) dBt dt + f 00(Bt) 2! . Multiply
defined by the SDE (23.8) is sometimes called the risk-neutral process
associated with the asset price process cfw_St , t 0 defined by (23.1). As
with the asset price process, the associated risk-neut
particular properties of these individual functions. Example 25.1.
Consider the function f(x) = x 2 . We see that the domain of f is all real
numbers, and the range of f is all non-negative real numbe
Example 15.4. If we combine our result of Example 14.5, namely Z t 0 B
3 s dBs = 1 4 B 4 t 3 2 Z t 0 B 2 s ds, with our result of Example 15.2,
namely Z t 0 sBs dBs = 1 2 tB2 t t 2 2 Z t 0 B 2 s ds ,
the defining SDE. Suppose that we are interested in determining the fair
price at time t = 0 of a European call option on the asset price cfw_St , t 0
with strike price E and expiry date T assuming a
T, x R, (23.5) subject to the terminal condition V (T, x) = (x). The
generalized Feynman-Kac representation theorem tells us that the
solution to b(t, x)g 0 (t, x) + 1 2 a 2 (t, x)g 00(t, x) + g(t, x)
derivatives in the space variable (the x-variable) and the dot ( ) notation
for derivatives in the time variable (the t-variable). That is, f 0 (t, x) =
xf(t, x), f00(t, x) = 2 x2 f(t, x), f(t, x) =
a number A of assets which is self-financing: d(t, St) = A(t, St) dSt +
rD(t, St) dt. As in Lecture #16, this implies that the change in V (t, St)
(t, St) over any time step is non-random and must eq
result for this integral. Example 14.3. Let f(x) = x 3 so that f 0 (x) = 3x 2
and f 00(x) = 6x. Therefore, Itos formula implies B 3 t B 3 0 = Z t 0 3B
2 s dBs + 1 2 Z t 0 6Bs ds so that rearranging yi
functions on R. Define the functional D by setting D(f) = f 0 for f X .
That is, f X 7 f 0 . Formally, we define D by (Df)(x) = f 0 (x) for every
x R, f X . Example 25.4. We have already seen a number
and that if a solution exists, then it must be unique. The no arbitrage
assumption (i.e., the put-call parity) implies that the condition (19.2)
always holds. We now calculate V 00() = 2V 2 = d1d2 V =
course, there are assumptions needed to ensure that Newtons method
converges and produces the correct solution. If we now consider F() =
V () V , then we have already shown that the conditions needed
B2 (t) + C with A = a i, B = 2 2 , C = i 2 + 2 2 . (22.4) This
ordinary differential equation can be solved by integration; see Exercise
22.1 below. The solution is given by (t) = D + E tan(F t + G) w
sum of angles identity for cosine therefore gives cos arctan D E F(T
t) = cos arctan D E cos (F(T t) + sin arctan D E sin (F(T t) = E
D2 + E2 cos (F(T t) D D2 + E2 sin (F(T t) = E cos (F(T t) D
sin
second order term vanishes as t 0 is not valid. Indeed, if we take
Itos Formula (Part I) 57 g(t) = Bt and divide by t, then we find f(Bt)
t = f(Bt+t) f(Bt) t = f 0 (Bt) Bt t + f 00(Bt) 2! (Bt) 2 t + f
+ (b Xt) dt (20.5) where and b are constants and cfw_Bt , t 0 is a
standard Brownian motion. The trick for solving the mean-reverting
Ornstein-Uhlenbeck process is similar. That is, we multiply by e t
up. The parameter a measures the speed of the mean-reversion, b is
the average level of volatility, and is the volatility of volatility. Market
data suggests that the correlation rate is typically neg